4.2. Sources of Possible Systematic Uncertainty

The absolute calibration of both the SZE and X-ray observations directly affects the distance determinations. In addition to the absolute calibration uncertainty from the observations, there are possible sources of systematic uncertainty that depend on the physical state of the ICM and other sources that can contaminate the cluster SZE emission. Table 1 summarizes the systematic uncertainties in the Hubble constant determined from 30 GHz interferometric SZE observations of a sample of 18 clusters (Reese et al., 2002), but are typical of most SZE experiments. The entries marked with asterisks are expected to average out for a large sample of clusters and include a 1 / 18^1/2 factor reflecting the 18 clusters used in this work. For detailed discussions of systematics see Birkinshaw (1999) and Reese et al. (2002); Reese et al. (2000).

4.2.1. Cluster Structure

Most clusters do not appear circular in radio, X-ray, or optical wavelengths. Under the assumption of axisymmetric clusters, the combined effect of cluster asphericity and its orientation on the sky conspires to introduce a roughly ± 20% random uncertainty in H₀ determined from one galaxy cluster (Hughes & Birkinshaw, 1998). When one considers a large, unbiased sample of clusters, with random orientations, the errors due to imposing a spherical model are expected to cancel, resulting in a precise determination of H₀. Numerical simulations using triaxial models support this assumption (Sulkanen, 1999).

Departures from isothermality in the cluster atmosphere may result in a large error in the distance determination from an isothermal analysis. The ROSAT band is fairly insensitive to temperature variations, showing a ~ 10% change in the PSPC count rate for a factor of 2 change in temperature for T_e > 1.5 keV gas (Mohr et al., 1999). A mixture of simulations and studies of nearby clusters suggests a 10% effect on the Hubble parameter due to departures from isothermality (e.g., Inagaki et al., 1995; Roettiger et al., 1997).

Clumping of the intracluster gas is a potentially serious source of systematic error in the determination of the Hubble constant. Unresolved clumps in an isothermal intracluster plasma will enhance the X-ray emission by a factor C², where

(7)

If significant substructure exists in galaxy clusters, the cluster generates more X-ray emission than expected from a uniform ICM, leading to an underestimate of the angular-diameter distance (D_A S_x0^-1) and therefore an overestimate of the Hubble parameter by a factor C². Unlike orientation bias, which averages down for a large sample of clusters, clumping must be measured in each cluster or estimated for an average cluster. There is currently no observational evidence of significant clumping in galaxy clusters. If clumping were significant and had large variations from cluster to cluster, we might expect larger scatter than is seen in the Hubble diagrams from SZE and X-ray distances (Fig. 6). In addition, the agreement between SZE (e.g., Grego et al., 2001) and X-ray (e.g., Mohr et al., 1999) determined gas fractions from galaxy clusters also suggests that clumping is not a large effect.

4.2.2. Possible SZE Contaminants

Undetected point sources near the cluster mask the central decrement, causing an underestimate in the magnitude of the decrement and therefore an underestimate of the angular diameter distance. Point sources in reference fields and for interferometers, the complicated synthesized beam shapes, may cause overestimates of the angular diameter distance. Massize clusters typically have central dominant (cD) galaxies, which are often radio bright. Therefore, it is likely that there is a radio point source near the center of each cluster. Typical radio sources have falling spectra, roughly 1, where S ^-. At 30 GHz, possible undetected point sources just below the detection threshold of the observations introduce a ~ 10% uncertainty.

Cluster peculiar velocities with respect to the CMB introduce an additional CMB spectral distortion known as the kinetic SZE (see Section 2.2). For a single isothermal cluster, the ratio of the kinetic SZE to the thermal SZE is

(8)

where _pec is the peculiar velocity along the line of sight. At low frequencies (Rayleigh-Jeans regime), f (x) -2 and the kinetic SZE is ~ 10% of the thermal SZE. Recent observational evidence suggests a typical one-dimensional rms peculiar velocity of ~ 300 km s^-1 (Watkins, 1997), and recent simulations found similar results (Colberg et al., 2000). In general, the kinetic effect is 10% that of the thermal SZE, except near the thermal null at ~ 218 GHz where the kinetic SZE dominates (see Fig. 2). Cluster peculiar velocities are randomly distributed, so when averaged over an ensemble of clusters, the effect from peculiar velocities should cancel.

CMB primary anisotropies have the same spectral signature as the kinetic SZE. The effects of primary anisotropies on cluster distances depend strongly on the beam size of the SZE observations and the typical angular scale of the clusters being observed (nearby versus distant clusters); the CMB effects on the inferred Hubble constant should average out over an ensemble of clusters. Recent BIMA observations provide limits on primary anisotropies on scales of a few arcminutes (Dawson et al., 2001; Holzapfel et al., 2000). On these scales, CMB primary anisotropies are an unimportant ( 1%) source of uncertainty. For nearby clusters, or for searches for distant clusters using beams larger than a few arcminutes, the intrinsic CMB anisotropy must be considered. The unique spectral signature of the thermal SZE can be used to separate it from primary CMB anisotropy. However, it will not be possible to separate primary CMB anisotropies from the kinetic SZE without relying on the very small spectral distortions of the kinetic SZE due to relativistic effects.

The SZE may be masked by large-scale diffuse nonthermal radio emission in clusters of galaxies, known as radio halos. If present, radio halos are located at the cluster centers, have sizes typical of galaxy clusters, and have a steep radio spectrum ( 1 - 3; Hanish 1982; Moffet & Birkinshaw 1989; Giovannini et al. 1999; Kempner & Sarazin 2001). Because halos are rare, little is known about their nature and origin, but they are thought to be produced by synchrotron emission from an accelerated or reaccelerated population of relativistic electrons (e.g., Schlickeiser et al., 1987; Jaffe, 1977; Roland, 1981; Dennison, 1980). Conservative and simplistic modeling of the possible effects of these halos implies a ~ 3% overestimate on the inferred Hubble parameter from radio halos (Reese et al., 2002). Reese et al. (2002) show Very Large Array NVSS contours overlaid on 30 GHz interferometric SZE images, suggesting that radio halos have little impact on SZE observations and therefore on SZE/X-ray distances.

Imprecisely measured beam shapes affect the inferred central decrements and therefore affect the Hubble constant. For interferometric observations, the primary beam is determined from holography measurements. Conservative and simple modeling of the effects of the primary beam suggests that the effect on the Hubble constant is a few percent ( 3%) at most. In theory, this is a controllable systematic with detailed measurements of beam shape and is currently swamped by larger potential sources of systematic uncertainty.